Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set $\Delta$, or alternatively, $X$ is isometric to a closed subspace of $C([0,1])$.

Having checked a few online resources, I have come to understand that not every such isometric embedding of a banach space $X$ will be complemented (i.e. possess a closed complement) in $C(\Delta)$ (or $C([0,1])$). In fact, Banach and Mazur themselves showed that any isometric copy of $l^1 \subset C([0,1])$ is noncomplemented.

I understand that this question might have a long answer in the literature, but here it goes: is there any kind of simple sufficient condition for a closed subspace of $C([0,1])$ to be complemented?

Failing at that, I'm also interested in isomorphic copies of $X$ in $C([0,1])$ where the norm of the isomorphism and its inverse are controlled in some way. For instance, what properties of $X$ guarantee the following: